Conditional convergence

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

Definition

More precisely, a series ${\textstyle \sum _{n=0}^{\infty }a_{n}}$  is said to converge conditionally if ${\textstyle \lim _{m\rightarrow \infty }\,\sum _{n=0}^{m}a_{n}}$  exists and is a finite number (not ∞ or −∞), but ${\textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=\infty .}$

A classic example is the alternating series given by

$1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n},$

which converges to $\ln(2)$ , but is not absolutely convergent (see Harmonic series).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in Rn can converge.

A typical conditionally convergent integral is that on the non-negative real axis of ${\textstyle \sin(x^{2})}$  (see Fresnel integral).